2.1 Level, Weight, Nominator and Denominator of an Eta Product. By an eta product we understand any finite product of functions. f(z) = m.
|
|
- Ethan Sims
- 6 years ago
- Views:
Transcription
1 Ea Producs.1 Level, Weigh, Noinaor and Denoinaor of an Ea Produc By an ea produc we undersand any finie produc of funcions f(z = η(z a where runs hrough a finie se of posiive inegers and he exponens a ay ake any values fro Z, posiive or negaive or 0. (Of course, an exponen 0 conribues a rivial facor 1 o he produc, and herefore we ay as well assue ha a 0forall. Since he produc is finie, he lowes coon uliple N =lc{} exiss, and every divides N. We wrie f(z = η(z a, (.1 and we call f an ea produc of level N. Here, forally, runs hrough all posiive divisors of he posiive ineger N, and soe of he exponens a igh be 0. We will use his noaion also in cases when N is bigger han lc{}; henn is a uliple of he level of he ea produc. Soe auhors use he er ea quoien for funcions as in (.1, and hey reserve he er ea produc for he case when a 0 for all. Ofen we will use he noaion [1 a1, a, 3 a3,...] as an abbreviaion for he ea produc η(z a1 η(z a η(3z a3... This noaion is adoped fro [4]. The er in square brackes will ofen be wrien as a fracion wih posiive exponens in is nueraor and denoinaor. G. Köhler, Ea Producs and Thea Series Ideniies, Springer Monographs in Maheaics, DOI / , c Springer-Verlag Berlin Heidelberg
2 3. Ea Producs An ea produc (.1 ransfors like a odular for of weigh k = 1 a wih soe uliplier syse on he congruence group Γ 0 (N. This eans ha for every L = ( a b c d Γ0 (N wehave ( az + b f(lz =f = v f (L(cz + d k f(z cz + d where v f (L is soe 4h roo of uniy which can be copued fro he uliplier syse v η of he ea funcion. We will rarely need o know he values v f (L of he uliplier syse of f explicily. We have ( a b v f (L =v f = ( ( a a b v c d η c/ d where he values of v η are given explicily in Theore 1.7. Highly iporan for us, however, is he value v f (T for he ranslaion T = ( We wrie 1 a = s (. 4 in lowes ers, i.e., wih gcd(s, = 1. Then i is a rivial consequence fro η(z +1=e ( 1 4 η(z hawehavevf (T =e ( s, ( s f(tz=f(z +1=e f(z. I follows ha f has a Fourier expansion of he for ( nz f(z = c n e n s(od, n s (.3 s wih coefficiens c n Z, c s = 1. In paricular, is he order of f a he cusp. We call s he nueraor and he denoinaor of he ea produc (.1. The denoinaor is a divisor of 4. An explici forula for v f (L is given in [105], Theore 1.64 in he case when he weigh k and he nuber (. are inegers (whence = 1 and when also a N/ is an ineger; in his case v f (L is a funcion of d only. 1 4 For a Fourier series (.3, he sign ransfor is f ( z + 1 = e ( s n s(od,n s ( 1 (n s/ c n e ( nz.
3 .. Ea Producs on he Fricke Group 33 Modifying our concep fro Sec. 1., we will also call he series for e( s f(z + 1 hesign ransfor of he series for f(z. An ea produc f of level N as in (.1 will be called old if here is an ineger d 1, a proper divisor N 1 of N and an ea produc g of level N 1 such ha f(z =g(dz. Oherwise f will be called a new ea produc. Since f and g have idenical Fourier coefficiens, i ofen suffices o sudy new ea producs. Neverheless, soeies i is advanageous o consider old ones. For exaple, g(z =η(zη(z andf(z =η(8zη(16z boh are old ea producs of level 16, while g is new of level. Bu f has period 1, and hence is Fourier expansion is a power series in he variable q = e(z, which igh be nicer han he expansion of g wih fracional powers of q. We ephasize ha our concep of a new ea produc has lile o do wih he concep of a newfor in he heory of Hecke operaors as explained in Sec Only occasionally i will happen ha a new ea produc is also a Hecke eigenfor. (Incidenally, η(zη(z is such an exaple; see Sec Ea Producs on he Fricke Group For he oen, le us pu f (z =η(z, where is a posiive ineger. Fro η( 1/z = iz η(z i follows ha ( f (W N z=f 1 ( 1 = η = ( N (in/zη Nz (N/z z. Thus, for an ea produc f of level N as in (.1, we obain f(w N z = ( ( a N ( i(n/z 1/ η z = ( ( iz 1/ η(z a N/ ( 1/ = ( iz k a N/ η(z a N/. The ea produc f ransfors like a odular for of weigh k for he Fricke group Γ (N if and only if f(w N z= ( i Nz k f(z. We see ha his holds if and only if he condiion a N/ = a for all (.4 is saisfied. An ea produc wih his propery will be called an ea produc on he Fricke group of level N.
4 34. Ea Producs We observe ha an ea produc of level N is deerined by is syse of τ(n exponens a, whereas roughly half of hese paraeers exacly τ(n/ of he suffice o deerine an ea produc on he Fricke group. Here, τ(n = σ 0 (N is he nuber of posiive divisors of N, as inroduced in Sec Expansion and Order a Cusps The produc for η(z ells us ha his funcion is nowhere 0. Therefore, ea producs (.1 are holoorphic on he upper half plane regardless of heir syse of exponens a. However, we will resric our sudy o ea producs which are holoorphic a all cusps, oo. In paricular, he order a he cusp should be non-negaive, i.e., s 0. We need condiions for an ea produc o be holoorphic a he oher cusps r Q. For his purpose we give a forula for he order of funcions η(z a an arbirary cusp and, soewha ore general, for he Fourier expansion of η(z a cusps. This expansion will evenually be useful when we wan o decide wheher a linear cobinaion of ea producs is a cusp for, where he ea producs are holoorphic a all cusps, bu no cusp fors heselves. Proposiion.1 Le f (z =η(z wih N, andler = d c Q be a reduced fracion wih c 0. Le a, b be chosen such ha A = ( a b SL (Z. Then we have: c d (1 The expansion of f a he cusp r is ( gcd(c, f (A 1 z = v η (L n=1 ( 1 n e ( n 1/ ( cz + a 4 ( (gcd(c, z + ν gcd(c, where L = ( x u SL (Z, x = d gcd(c,, u = c gcd(c,,andν is soe ineger. ( The order of f a he cusp r is ord(f,r= 1 4 (gcd(c,.
5 .3. Expansion and Order a Cusps 35 Proof. Since c, d are relaively prie, we can choose a, b Z such ha A = ( a b c d SL (Z. We ge A 1 ( = ( d b c a ( = d c = r and ( dz b f (A 1 z=η = η(αz cz + d where α = ( d b c a,de(α =. The expansion of f a r is given by he expansion of f (A 1 za. In order o find i, we need soe arix L = ( x y u v SL (Z such ha he lower lef enry in L 1 α vanishes. We have ( ( ( L 1 v y d b α = =. u x c a du cx Therefore we need ha du + cx = 0. Thus for he firs colun of L we can choose he relaively prie inegers x = d gcd(c, d = d g, u = c, wih g = gcd(c,. g Fro de(l 1 α=de(α = we infer ha ( ( L 1 g ν α = = 0 /g 0 /g wih soe ν Z. (Observe ha we can copue ν = bv ya explicily, depending on and r. Now we ge f (A 1 z = η(αz =η(ll 1 αz ( = v η (L u gz + ν 1/ /g + v η(l 1 αz ( 1/ ( cz cν/g gz + ν = v η (L + v η /g /g ( ( 1/ g cν v g = v η (L cz η( g z + νg ( 1/ g g = v η (L ( cz + a η( z + νg ( 1/ g ( ( 1 n = v η (L ( cz + a e n 4 (g z + νg. This proves our firs asserion. The firs non-vanishing er in ( cz+a 1/ f (A 1 z is a consan uliple of e(g z/4. Thus, by our definiion of he order, we obain ord(f,r=g /4, which is he second asserion. n=1 We noe an iediae consequence of he second asserion:
6 36. Ea Producs Corollary. Le f be an ea produc as in (.1, andler = d c Q, gcd(c, d =1. Then he order of f a he cusp r is ord(f,r = 1 (gcd(c, a. 4 An ea produc f will be called a holoorphic ea produc if is orders a all cusps are non-negaive, ord(f,r 0 for all r Q. Holoorphic ea producs (.1 are (enire odular fors for Γ 0 (N. They are cusp fors if and only if all he orders are posiive, ord(f,r > 0 for all r Q. In his case we will call he cuspidal ea producs, andnon-cuspidal oherwise..4 Condiions for Holoorphic Ea Producs Fro Corollary. we ge condiions for an ea produc o be holoorphic or a cusp for. These are condiions for infiniely any cusps. Of course, i suffices o check hese condiions for a finie syse of represenaives of inequivalen cusps of Γ 0 (N, i.e., for he orbis of his group on Q.The nuber of inequivalen cusps of Γ 0 (N is ϕ(gcd(, N/, where ϕ is he Euler funcion; his is known fro several exbooks; see [15], p. 10, for exaple. A se of represenaives of inequivalen cusps is given in [9], forula (. Using his, i would be possible o characerize holoorphic and cuspidal ea producs by syses of finiely any inequaliies. In fac, one can find such a characerizaion using nohing else bu Corollary.: We observe ha he order of f a a cusp does only depend on he denoinaor c of ha cusp. If is any divisor of N hen for all c Z we have gcd(c, = gcd(gcd(c, N,, and gcd(c, N isadivisorofn. Therefore he condiions ord(f,r 0are saisfied for all r Q if and only if ord(f,1/c 0 for all c N, and siilarly for sric inequaliies. This proves he following resul:
7 .5. The Cones and Siplices of Holoorphic Ea 37 Corollary.3 An ea produc f as in (.1 is holoorphic if and only if he inequaliies (gcd(c, a 0 hold for all posiive divisors c of N. I is a cuspidal ea produc if and only if all hese inequaliies hold sricly..5 The Cones and Siplices of Holoorphic Ea Producs According o Corollary.3, we inroduce raional nubers α c, a arix A and a colun vecor X by (gcd(c, α c =, A = A(N =(α c c,, X =(a R τ(n, (.5 where he posiive divisors, c of N are aken in soe arbirary, bu fixed order. (Usually he divisors will be in heir naural order. Then he condiion for holoorphic ea producs of level N reads A(N X 0, (.6 and cuspidal ea producs are characerized by A(N X>0. The syse of linear inequaliies in (.6 defines an inersecion of τ(n closed halfspaces in R τ(n whose bounding hyperplanes all pass hrough he origin. So his syse defines a closed siplicial cone wih is verex a he origin. We denoe his cone by K(N, i.e. K(N ={X R τ(n A(NX 0}. (.7 We can reforulae Corollary.3 as follows: Corollary.4 An ea produc (.1 is holoorphic if and only if is vecor of exponens X =(a is a laice poin in he cone K(N. I is cuspidal if and only if X is an inerior poin of K(N.
8 hp://
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationFourier Series & The Fourier Transform. Joseph Fourier, our hero. Lord Kelvin on Fourier s theorem. What do we want from the Fourier Transform?
ourier Series & The ourier Transfor Wha is he ourier Transfor? Wha do we wan fro he ourier Transfor? We desire a easure of he frequencies presen in a wave. This will lead o a definiion of he er, he specru.
More informationProblem set 2 for the course on. Markov chains and mixing times
J. Seif T. Hirscher Soluions o Proble se for he course on Markov chains and ixing ies February 7, 04 Exercise 7 (Reversible chains). (i) Assue ha we have a Markov chain wih ransiion arix P, such ha here
More informationREMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS 1. INTRODUCTION
REMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS YUK-KAM LAU, YINGNAN WANG, DEYU ZHANG ABSTRACT. Le a(n) be he Fourier coefficien of a holomorphic cusp form on some discree subgroup
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationRiemann Hypothesis and Primorial Number. Choe Ryong Gil
Rieann Hyohesis Priorial Nuber Choe Ryong Gil Dearen of Maheaics Universiy of Sciences Gwahak- dong Unjong Disric Pyongyang DPRKorea Eail; ryonggilchoe@sar-conek Augus 8 5 Absrac; In his aer we consider
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationGCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS
GCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS D. D. ANDERSON, SHUZO IZUMI, YASUO OHNO, AND MANABU OZAKI Absrac. Le A 1,..., A n n 2 be ideals of a commuaive ring R. Le Gk resp., Lk denoe
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationWave Mechanics. January 16, 2017
Wave Mechanics January 6, 7 The ie-dependen Schrödinger equaion We have seen how he ie-dependen Schrodinger equaion, Ψ + Ψ i Ψ follows as a non-relaivisic version of he Klein-Gordon equaion. In wave echanics,
More informationQuestion 1: Question 2: Topology Exercise Sheet 3
Topology Exercise Shee 3 Prof. Dr. Alessandro Siso Due o 14 March Quesions 1 and 6 are more concepual and should have prioriy. Quesions 4 and 5 admi a relaively shor soluion. Quesion 7 is harder, and you
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationNEW EXAMPLES OF CONVOLUTIONS AND NON-COMMUTATIVE CENTRAL LIMIT THEOREMS
QUANTUM PROBABILITY BANACH CENTER PUBLICATIONS, VOLUME 43 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 998 NEW EXAMPLES OF CONVOLUTIONS AND NON-COMMUTATIVE CENTRAL LIMIT THEOREMS MAREK
More informationVOL. 1, NO. 8, November 2011 ISSN ARPN Journal of Systems and Software AJSS Journal. All rights reserved
VOL., NO. 8, Noveber 0 ISSN -9833 ARPN Journal of Syses and Sofware 009-0 AJSS Journal. All righs reserved hp://www.scienific-journals.org Soe Fixed Poin Theores on Expansion Type Maps in Inuiionisic Fuzzy
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
More informationOn the approximation of particular solution of nonhomogeneous linear differential equation with Legendre series
The Journal of Applied Science Vol. 5 No. : -9 [6] วารสารว ทยาศาสตร ประย กต doi:.446/j.appsci.6.8. ISSN 53-785 Prined in Thailand Research Aricle On he approxiaion of paricular soluion of nonhoogeneous
More informationU( θ, θ), U(θ 1/2, θ + 1/2) and Cauchy (θ) are not exponential families. (The proofs are not easy and require measure theory. See the references.
Lecure 5 Exponenial Families Exponenial families, also called Koopman-Darmois families, include a quie number of well known disribuions. Many nice properies enjoyed by exponenial families allow us o provide
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationHeat kernel and Harnack inequality on Riemannian manifolds
Hea kernel and Harnack inequaliy on Riemannian manifolds Alexander Grigor yan UHK 11/02/2014 onens 1 Laplace operaor and hea kernel 1 2 Uniform Faber-Krahn inequaliy 3 3 Gaussian upper bounds 4 4 ean-value
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationLet ( α, β be the eigenvector associated with the eigenvalue λ i
ENGI 940 4.05 - Sabiliy Analysis (Linear) Page 4.5 Le ( α, be he eigenvecor associaed wih he eigenvalue λ i of he coefficien i i) marix A Le c, c be arbirary consans. a b c d Case of real, disinc, negaive
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationChapter 9 Sinusoidal Steady State Analysis
Chaper 9 Sinusoidal Seady Sae Analysis 9.-9. The Sinusoidal Source and Response 9.3 The Phasor 9.4 pedances of Passive Eleens 9.5-9.9 Circui Analysis Techniques in he Frequency Doain 9.0-9. The Transforer
More informationLogarithmic limit sets of real semi-algebraic sets
Ahead of Prin DOI 10.1515 / advgeom-2012-0020 Advances in Geomery c de Gruyer 20xx Logarihmic limi ses of real semi-algebraic ses Daniele Alessandrini (Communicaed by C. Scheiderer) Absrac. This paper
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationSections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationOn Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More informationA Generalization of Student s t-distribution from the Viewpoint of Special Functions
A Generalizaion of Suden s -disribuion fro he Viewpoin of Special Funcions WOLFRAM KOEPF and MOHAMMAD MASJED-JAMEI Deparen of Maheaics, Universiy of Kassel, Heinrich-Ple-Sr. 4, D-343 Kassel, Gerany Deparen
More informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationTHE FINITE HAUSDORFF AND FRACTAL DIMENSIONS OF THE GLOBAL ATTRACTOR FOR A CLASS KIRCHHOFF-TYPE EQUATIONS
European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN 59-995 HE FINIE HAUSDORFF AND FRACAL DIMENSIONS OF HE GLOBAL ARACOR FOR A CLASS KIRCHHOFF-YPE EQUAIONS Guoguang Lin & Xiangshuang Xia Deparen
More informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationINDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE
INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE JAMES ALEXANDER, JONATHAN CUTLER, AND TIM MINK Absrac The enumeraion of independen ses in graphs wih various resricions has been a opic of much ineres
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationEssential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems
Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor
More informationLecture 23 Damped Motion
Differenial Equaions (MTH40) Lecure Daped Moion In he previous lecure, we discussed he free haronic oion ha assues no rearding forces acing on he oving ass. However No rearding forces acing on he oving
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationRANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY
ECO 504 Spring 2006 Chris Sims RANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY 1. INTRODUCTION Lagrange muliplier mehods are sandard fare in elemenary calculus courses, and hey play a cenral role in economic
More informationnon -negative cone Population dynamics motivates the study of linear models whose coefficient matrices are non-negative or positive.
LECTURE 3 Linear/Nonnegaive Marix Models x ( = Px ( A= m m marix, x= m vecor Linear sysems of difference equaions arise in several difference conexs: Linear approximaions (linearizaion Perurbaion analysis
More informationThe minimum number of nonnegative edges in hypergraphs
The minimum number of nonnegaive edges in hypergraphs Hao Huang DIMACS Rugers Universiy New Brunswic, USA huanghao@mahiasedu Benny Sudaov Deparmen of Mahemaics ETH 8092 Zurich, Swizerland benjaminsudaov@mahehzch
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationOn the cohomology groups of certain quotients of products of upper half planes and upper half spaces
On he cohomolog groups of cerain quoiens of producs of upper half planes and upper half spaces Amod Agashe and Ldia Eldredge Absrac A heorem of Masushima-Shimura shows ha he he space of harmonic differenial
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationt 2 B F x,t n dsdt t u x,t dxdt
Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.
More informationNote on oscillation conditions for first-order delay differential equations
Elecronic Journal of Qualiaive Theory of Differenial Equaions 2016, No. 2, 1 10; doi: 10.14232/ejqde.2016.1.2 hp://www.ah.u-szeged.hu/ejqde/ Noe on oscillaion condiions for firs-order delay differenial
More informationM x t = K x F t x t = A x M 1 F t. M x t = K x cos t G 0. x t = A x cos t F 0
Forced oscillaions (sill undaped): If he forcing is sinusoidal, M = K F = A M F M = K cos G wih F = M G = A cos F Fro he fundaenal heore for linear ransforaions we now ha he general soluion o his inhoogeneous
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationMonochromatic Infinite Sumsets
Monochromaic Infinie Sumses Imre Leader Paul A. Russell July 25, 2017 Absrac WeshowhahereisaraionalvecorspaceV suchha,whenever V is finiely coloured, here is an infinie se X whose sumse X+X is monochromaic.
More informationOn the Infinitude of Covering Systems with Least Modulus Equal to 2
Annals of Pure and Applied Mahemaics Vol. 4, No. 2, 207, 307-32 ISSN: 2279-087X (P), 2279-0888(online) Published on 23 Sepember 207 www.researchmahsci.org DOI: hp://dx.doi.org/0.22457/apam.v4n2a3 Annals
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationTHE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX
J Korean Mah Soc 45 008, No, pp 479 49 THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Reprined from he Journal of he
More informationTHE MATRIX-TREE THEOREM
THE MATRIX-TREE THEOREM 1 The Marix-Tree Theorem. The Marix-Tree Theorem is a formula for he number of spanning rees of a graph in erms of he deerminan of a cerain marix. We begin wih he necessary graph-heoreical
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationPractice Problems - Week #4 Higher-Order DEs, Applications Solutions
Pracice Probles - Wee #4 Higher-Orer DEs, Applicaions Soluions 1. Solve he iniial value proble where y y = 0, y0 = 0, y 0 = 1, an y 0 =. r r = rr 1 = rr 1r + 1, so he general soluion is C 1 + C e x + C
More informationOn a Class of Two Dimensional Twisted q-tangent Numbers and Polynomials
Inernaiona Maheaica Foru, Vo 1, 17, no 14, 667-675 HIKARI Ld, www-hikarico hps://doiorg/11988/if177647 On a Cass of wo Diensiona wised -angen Nubers and Poynoias C S Ryoo Deparen of Maheaics, Hanna Universiy,
More informationWavelet Methods for Time Series Analysis. What is a Wavelet? Part I: Introduction to Wavelets and Wavelet Transforms. sines & cosines are big waves
Wavele Mehods for Time Series Analysis Par I: Inroducion o Waveles and Wavele Transforms waveles are analysis ools for ime series and images as a subjec, waveles are relaively new (983 o presen) a synhesis
More informationAfter the completion of this section the student. Theory of Linear Systems of ODEs. Autonomous Systems. Review Questions and Exercises
Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 V.5 SYSTEMS OF FIRST ORDER LINEAR ODEs Objecives: Afer he compleion of his secion he suden - should recall he definiion of a sysem of linear
More informationThus the force is proportional but opposite to the displacement away from equilibrium.
Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationA HARDY TYPE GENERAL INEQUALITY IN L p( ) (0, 1) WITH DECREASING EXPONENT
Transacions of NAS of Azerbaijan, 23, vol. XXXIII, No, pp. 45-5. 45 Farman I. MAMEDOV, Firana M. MAMEDOVA A HARDY TYPE GENERAL INEQUALITY IN L p ), ) WITH DECREASING EXPONENT Absrac We derive a Hardy ype
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationHomework sheet Exercises done during the lecture of March 12, 2014
EXERCISE SESSION 2A FOR THE COURSE GÉOMÉTRIE EUCLIDIENNE, NON EUCLIDIENNE ET PROJECTIVE MATTEO TOMMASINI Homework shee 3-4 - Exercises done during he lecure of March 2, 204 Exercise 2 Is i rue ha he parameerized
More informationAn Excursion into Set Theory using a Constructivist Approach
An Excursion ino Se Theory using a Consrucivis Approach Miderm Repor Nihil Pail under supervision of Ksenija Simic Fall 2005 Absrac Consrucive logic is an alernaive o he heory of classical logic ha draws
More informationThe following report makes use of the process from Chapter 2 in Dr. Cumming s thesis.
Zaleski 1 Joseph Zaleski Mah 451H Final Repor Conformal Mapping Mehods and ZST Hele Shaw Flow Inroducion The Hele Shaw problem has been sudied using linear sabiliy analysis and numerical mehods, bu a novel
More informationEXPONENTIAL PROBABILITY DISTRIBUTION
MTH/STA 56 EXPONENTIAL PROBABILITY DISTRIBUTION As discussed in Exaple (of Secion of Unifor Probabili Disribuion), in a Poisson process, evens are occurring independenl a rando and a a unifor rae per uni
More informationb denotes trend at time point t and it is sum of two
Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: 2454356X)
More informationWHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL
WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL DAVID M. HOWARD AND STEPHEN J. YOUNG Absrac. The linear discrepancy of a pose P is he leas k such ha here is a linear exension L of P such ha if x and y are incomparable,
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationPhysics for Scientists & Engineers 2
Direc Curren Physics for Scieniss & Engineers 2 Spring Semeser 2005 Lecure 16 This week we will sudy charges in moion Elecric charge moving from one region o anoher is called elecric curren Curren is all
More informationBBP-type formulas, in general bases, for arctangents of real numbers
Noes on Number Theory and Discree Mahemaics Vol. 19, 13, No. 3, 33 54 BBP-ype formulas, in general bases, for arcangens of real numbers Kunle Adegoke 1 and Olawanle Layeni 2 1 Deparmen of Physics, Obafemi
More informationSecond-Order Boundary Value Problems of Singular Type
JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 226, 4443 998 ARTICLE NO. AY98688 Seond-Order Boundary Value Probles of Singular Type Ravi P. Agarwal Deparen of Maheais, Naional Uniersiy of Singapore,
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More information